Small sets in convex geometry and formal independence over ZFC

Abstract

To each closed subset $S$ of a finite-dimensional Euclidean space corresponds a $\sigma$-ideal of sets $𝒥\left(S\right)$ which is $\sigma$-generated over $S$ by the convex subsets of $S$. The set-theoretic properties of this ideal hold geometric information about the set. We discuss the relation of reducibility between convexity ideals and the connections between convexity ideals and other types of ideals, such as the ideals which are generated over squares of Polish space by graphs and inverses of graphs of continuous self-maps, or Ramsey ideals, which are generated over Polish spaces by the homogeneous sets with respect to some continuous pair coloring. We also attempt to present to nonspecialists the set-theoretic methods for dealing with formal independence as a means of geometric investigations.